Abstract

We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertibility. As consequences, we deduce some global inversion and global injectivity results for Lipschitz mappings on Rn\mathbb{R}^n in terms of spectral conditions of its Clarke generalized differential.

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Jesús A. Jaramillo

Dep. de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

jaramil@mat.ucm.es

Óscar Madiedo

Dep. de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

oscar.reynaldo@mat.ucm.es

Luis Sánchez-González

Dep. de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

lfsanche@mat.ucm.es

Keywords

  • Global inversion
  • generalized differential
  • Finsler manifolds

Mathematics Subject Classification

49J52, 58C15

References

  1. D. Azagra, J. Ferrera, F. López-Mesas: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220(2) (2005) 304–361.https://doi.org/10.1016/j.jfa.2004.10.008Google Scholar
  2. C. Biasi, C. Gutiérrez, E. L. dos Santos: Global inverse mapping theorems, Cad. Mat. 10 (2009) 9–18.Google Scholar
  3. F. H. Clarke: Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975) 247–262.https://doi.org/10.1090/S0002-9947-1975-0367131-6Google Scholar
  4. F. H. Clarke: On the inverse function theorem, Pac. J. Math 64(1) (1976) 97–102.https://doi.org/10.2140/pjm.1976.64.97Google Scholar
  5. F. H. Clarke: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia (1990).https://doi.org/10.1137/1.9781611971309Google Scholar
  6. K. Deimling: Nonlinear Functional Analysis, Springer, New York (1985).https://doi.org/10.1007/978-3-662-00547-7Google Scholar
  7. A. Fernandes, C. Gutiérrez, R. Rabanal: On Local Diffeomorphisms of Rn that are Injective, Qual. Theory Dyn. Syst. 4(2) (2004) 255–262.https://doi.org/10.1007/BF02970861Google Scholar
  8. I. Garrido, O. Gutú, J. A. Jaramillo: Global inversion and covering maps on length spaces, Nonlinear Anal. 73(5) (2010) 1364–1374.https://doi.org/10.1016/j.na.2010.04.069Google Scholar
  9. O. Gutú, J. A. Jaramillo: Global homeomorphisms and covering projections on metric spaces, Math. Ann. 338(5) (2007) 75–95.https://doi.org/10.1007/s00208-006-0068-9Google Scholar
  10. J. Hadamard: Sur les transformations ponctuelles, Bull. Soc. Math. Fr. 34 (1906) 71–84.https://doi.org/10.24033/bsmf.771Google Scholar
  11. F. John: On quasi-isometric maps I, Commun. Pure Appl. Math. 21 (1968) 77–110.https://doi.org/10.1002/cpa.3160210107Google Scholar
  12. M. Jiménez-Sevilla, L. Sánchez-González: On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler manifolds, Nonlinear Anal. 74(11) (2011) 3487–3500.https://doi.org/10.1016/j.na.2011.03.004Google Scholar
  13. S. Lang: Fundamentals of Differential Geometry, Graduate Texts in Mathematics 191, Springer, New York (1999).https://doi.org/10.1007/978-1-4612-0541-8Google Scholar
  14. J. W. Milnor, J. D. Stasheff: Characteristic Classes, Princeton University Press, Princeton (1974).https://doi.org/10.1515/9781400881826Google Scholar
  15. R. S. Palais: Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966) 115–132.https://doi.org/10.1016/0040-9383(66)90013-9Google Scholar
  16. R. Plastock: Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974) 169–183.https://doi.org/10.1090/S0002-9947-1974-0356122-6Google Scholar
  17. B. H. Pourciau: Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl. 22(3) (1977) 311–351.https://doi.org/10.1007/BF00932859Google Scholar
  18. B. H. Pourciau: Global invertibility of nonsmooth mappings, J. Math. Anal. Appl. 131(1) (1988) 170–179.https://doi.org/10.1016/0022-247X(88)90198-9Google Scholar
  19. P. J. Rabier: Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. Math. 146(3) (1997) 647–691.https://doi.org/10.2307/2952457Google Scholar
  20. E. H. Spanier: Algebraic Topology, McGraw Hill, New York (1966).https://doi.org/10.1007/978-1-4684-9322-1_5Google Scholar

Suggested citation

J. A. Jaramillo, O. Madiedo, L. Sánchez-González. “Global Inversion of Nonsmooth Mappings on Finsler Manifolds.” Journal of Convex Analysis 20 (2013), No. 4, 1127–1146. https://doi.org/10.68381/jca20061

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.